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A 

Primer of Logic 






HENRY BRADFORD SMITH 

Assistant Professor of Philosophy in the 
Uni<versity of Pennsylnjania 



B. D. Smith A Bros. 

Pulaski. Va. 

1917 






I THE LIBRARY 
or CONORESt 

iwARMmoreit 






PREFACE 

In the pages which follow will be found the outlines 
of the logic of a set of categorical forms which are not 
Aristotle's own. It is only the fragment of a general 
theory, but the content of all the chapters of the old logic, 
which are commonly regarded as essential, except that 
one which deals with the calculus of classes, will be found 
included. In the first appendix the relation of these new 
forms to the traditional ones has been pointed out in detail. 

I have at least one debt, which calls for a definite 
acknowledgement. It is to my friend and teacher, Mr. 
Edgar A. Singer, Jr., that I owe what training I have had 
in the science. He has never failed, through hints thrown 
out in conversation, to correct my misapprehensions. 
But my indebtedness is more specific than this. I have so 
far employed his own method, a method developed in his 
academic lectures, that I could scarcely have ventured 
upon the publication of these outlines of a theory without 
his express permission. 

H. B. S. 



TABLE OF CONTENTS 

CHAPTER I PAGES 

§1-3. The fundamental properties of the categorical 

forms. Exercises 1-9 

CHAPTER n 

§4-8. The relationships of "better" and "worse." 
Deduction of the moods of immediate infer- 
ence and of the syllogism by rule. Exercises. . . 10-17 

CHAPTER ni 

§9-14. The relationships of "better" and "worse" 
defined. Symbolic deduction of the moods 
of immediate inference and of the syllogism. 
Exercises 18-27 

CHAPTER IV 
§15. The general solution of the sorites. Exercises.. .28-34 

APPENDIX I 

On the simplification of categorical expression 

and the reduction of the syllogistic figures 35-44 

APPENDIX II 

Historical note on De Morgan's new proposi- 
tional forms 45-48 



CHAPTER I 

§1. In 1846 Sir William Hamilton published the 
prospectus of an essay on a "New Analytic of Logical 
Forms,"* which revived the question as to whether or 
not the quantity of the predicate of the categorical forms 
should be stated explicitly. The chief difficulties of his 
system result from the ambiguity of the meaning of some, 
from the impossibility of making every form of categorical 
expression simply convertible, and from the seemingly 
curious effort to establish an order of better and worse 
between the relations connecting subject and predicate. 
Four of Hamilton's eight forms are redundant. The four 
that remain will be represented here by the letters, a, 

/5, 7, c. 

Accordingly let 

aab=all a is all b, 
/3ab =some a is some b, 
7ab =all a is some b, 
eab=no a is b. 
Here some, the some expressed explicitly in ^ and 7, 
means some at least, not all. This meaning of the word 
is established unambiguously by the properties of the 
forms. In addition to these abbreviations we will employ 
the notation: 

Xab =Xab (is true), 
x'ab=Xab (is false), 

Xab * yab =Xab (is truc) and yab (is true), 
Xab + yab =Xab (is truc) or yab (is true), 
Xab ^ yab =Xab (is truc) impHcs yab (is true), 
(xab ^ yab)' =Xab (is toie) docs not imply yab (is true). 

*Lectures on Logic, ed. by Mansel and Veitch, Boston, 
Gould and Lincoln, 1863. 



2 A PRIMER OF LOGIC 

§2. The implications, which are given below, express 
the chief characteristics of the forms. The theorems 
follow by the principle of the denial of the consequent, 
which may be written in the two forms: 

(x Z yO Z (y Z x') and (x' Z y)' Z (y' Z x)'. 

Postulates:"^ 

aab Z /3'ab iSab Z 7'ab (a'ab Z i^ab)' (i^'ab Z Tab)' 

Ctab Z 7'ab /3ab Z e'ab (ct'ab ^ Tab)' (^'ab Z €ab)' 

ttab Z e'ab Tab Z e'ab (tt'ab Z eab)' (T^ab Z eab)' 

Tab / T'ba (T'ab Z Tba)' 

Theorems: 

eab -^ a'ab Tab -^ « ab ( e'ab ^ Ctab) (T^ab Z ttab)' 

6ab Z ^'ab Tab ^ iS'ab ( e'ab Z i^ab)' (T^ab Z i^ab)' 

€ab Z T'ab /5ab Z a'ab (c'abZTab)' (^'ab Z ttab)' 

Let us postulate in addition that: 

ttab Z (a'ab)' (ct'ab)' ^ Ctab 

iSab / (^'ab)' (iS'ab)' Z i^ab 

Tab / (T'ab)' (T'ab)' Z Tab 

eabZ(€'ab)' (e'ab)'Z €ab 

Then, if kab=i=Wab and if kab and Wab represent only 
the unprimed letters, aab, i^ab, Tab, ^ab, a complete induction 
of the propositions given above yields the general result: 

kabZ (k'ab)', (k'ab)'Z kab, I 

kab Z W'ab, (W'ab Z kab)', II 

*These assumptions are in accord with those of the common 
logic, but no longer hold when the terms are allowed to take on 
the limiting values and 1 ; for Toi and eoi are both true prop- 
ositions. The assumptions (Tab Z e'ab)' and (eabZT'ab)' will 
be characteristic of a more general logic, which will include 
the classical logic as a special case. (See the concluding remarks 
of chapter III.) 



A PRIMER OF LOGIC 3 

and by denial of the consequent, 

(xZy) Z(y'ZxO, 
(xZy)'Z(y'ZxO', 

it follows that: 

(k'ab)'Zw'ab, (w'abZ (k'ab/)'- HI 

If now we postulate: 

(ttab ^ Ct'ab)' (a'ab ^ ^ab)' 

(^abZ^'ab)' (^'abZ^.b)' 

(Tab ^ T'ab)' (7'ab / Tab)' 

(eabZ e'ab)' (e'abZ €ab)' 

and, consequently, 

(kabZk'ab)', (k'abZkab)', 

it will follow by* 

(xZy) (xZz)'Z(yZz)', 
(xZz)'(yZz)Z(xZy)', 

that (kabZ (w'ab)O'. ((W'ab/Z kab)'. IV 

Finally if we assume 

Ctab ^ Ctba, ^ab ^ ^ba> Tab Z Tab, ^ab ^ €ba, 

it will follow, by 

(x Z y) (y Z z) Z (x Z z) and (x Z y) Z (y' Z x'), 
that 

kab ^ kab, k ab '<^ k ab« I 

Definitions, 

If Xab Z y'ab and y'ab ^ Xab, Xab is said to be contra- 
dictory to yab. By I, kab is contradictory to k'ab and, by 
I', k'ab is contradictory to kab- 

♦These are obtained from (x Z y) (y Z z) Z (x Z z) by 
(xy Z z) Z (xz' Z yO and xy Z yx. 



4 A PRIMER OF LOGIC 

If Xab ^ y'ab and (y'ab ^ Xab)', Xab IS Said to be contrary 
to Yab. By II, kab is contrary to Wab. and conversely, 
since kab and Wab are interchangeable. 

If (xab ^ y'ab)' and y'ab ^ Xab, Xab IS Said to be subcon- 
trary to yab- By III, k'ab and w'ab are subcontrary pairs. 

If (xab ^ y'ab)' and (y'ab Z Xab)', Xab is Said to be suh- 
alternate to yab- By IV, k'ab and Wab are subalternate 
pairs. 

§3. Having classified the categorical forms under 
these heads, it remains to differentiate them by means of 
their formal properties. If we assume as valid, 

Q- aa ■^ Ctaa, € aa ■^ ^afi, V.^aa •^ Ct aa/ j V ^afi ^ € afi/ » 

where a represents the class contradictory to a, (non-a), 
the other propositions given below may be derived.* 

a aa '^ ttaa V^taa •^ Ct aa/ Ct aa ^ 0- aa (,Ct aa ^ ^aa/ 

/3aa Z iS'aa (i^'aa Z /^aa)' /3 aa ^ /3'aa (/3'aa Z i^aa)' V 

7aa Z 7'aa (7'aa Z Taa)' 7 aa ^ 7'aa (7'aa ^ 7aa)' 

€aa -^ € aa C^aa^*^ ^&b.) ^ && ^ ^&& (Caa-^ C aa) 

The results of V, together with the non-convertible 
character of 7**, are enough to establish the definitions 
of the four forms. 



*Under the conditions mentioned above, in note, p. 2, we 
shall have to write (7aa / 7'aa)'. Implications V are an extension 
of the meaning of implication, made necessary by our having 
to call Qiaa and €aa true propositions. (See Boole, Investigation 
of the Laws of Thought, chap. XI, p. 169.) 

**The operation of simple conversion consists in interchang" 
ing subject and predicate. By the principle, (x Z z)'(y Z z) Z 
(xZ y)', and what has gone before, we have: 

(7ab^ 7'ab)'(7baZ 7'ab) ^ (7ab Z 7ba)'. 



A PRIMER OF LOGIC 5 

Definitions* 

A form which is the contrary of itself is called a null- 
form. By V, /3aa, 7aa, ^aa, oL&h, i^aa, 7afi, are null-forms. 

A form which is the subcontrary of itself is called a 
one-form. By V, aaa, €aa, are one-forms. 

If Xab is unprimed and Xaa a one-form then Xab is 
called an a-form. 

If Xab is unprimed and simply convertible and if 
Xaa and Xaa arc null-forms, then Xab is called a jS-form. 

If Xab is unprimed and not simply convertible, then 
Xab is called a 7-form. 

If Xab is unprimed and Xaa a one-form, then Xab is 
called an e-form. 



A PRIMER OF LOGIC 



EXERCISES 



(1) Assuming kab = kab * kab, kab^ w'ab, show that, 

«ab ^ P ab 7 ab C ab 7 ba 
/3ab -<^ « ab 7 ab e ab 7 ba 
7ab -^ « ab /5 ab e ab 7 ba 

€ab ^ « ab /3 ab 7 ab 7 ba 

by the aid of, 

(xZ y) (yZ z)Z (xZ z), 
(xZ y) Z (zxZ zy). 

Equality is defined by 

(xZy)(yZx)Z(x = y). 
(x = y)Z(xZy) (yZx). 

If now 

oi II I / 

P ab 7 ab € ab 7 ba Z QJab 

/ / / I/O 

Oi ab 7^ab € ab 7 ba Z p^\, 

« ab /5 ab € ab 7 ba Z 7ab 
a ab /5 ab 7 ab 7 ba Z €ab 

it will follow that 

«ab = |3 ab 7 ab € ab 7 ba « ab =^ab + 7ab + €ab + 7ba 

jSab =«'ab 7'ab €'ab 7^ba /3^ab =«ab + 7ab + €ab + 7ba 

7ab =« ab /3 ab € ab 7 ba 7 ab =«ab + ^ab + €ab + 7ba 

€ab =« ab ^ ab 7 ab 7 ba € ab =«ab + /3ab + 7ab + 7ba 

The second set of equations follows from the first by the 
principle, that the contradictory of a product is the sum of the 
contradictories of the separate factors, and by substituting kab 
directly for (k'ab)'. 

(2) Show by the method of the last example that 

«ab=«ab ^ ab=«ab 7 ab=«ab € ab 
/3ab =^ab a ab = /^ab 7 ab = ^ab € ab 
7ab =7ab Oi ab =7ab ^ ab = 7ab € ab 
€ab = €ab « ab = €ab /3 ab = ^ab 7 ab 

Show too that 

«ab=«ab i^'ab 7'ab=«ab ^'ab t'ab=aab7'ab e'ab. CtC, CtC. 

and that aab=aab i^'ab 7'ab e'ab» etc., etc. 
Derive the analogues of the first set: 

a'ab = «'ab + ^ab =«'ab + 7ab =a'ab + eab. etC, CtC. 



A PRiMER OF LOGIC 1 

Accordingly, since kab = kab * w'ab and k'ab = k'ab + Wab, any 
primed letter is a modulus of multiplication with respect to 
any unprimed letter not itself and any unprimed lett r is a 
modulus of addition with respect to any primed letter not itself. 

The propositional zero is defined by 

(xyZ (xy)OZ(xyZ 0), 
(xyZ 0)Z(xyZ (xy)0, 

and the propositional one by 

((xy/Z xy)Z(lZ xy), 
(lZxy)Z((xy/Z xy). 

From the principles, 

(xZ yOZ(xyZ (xy)'), 
(xyZ (xy)0 Z(xZ yO, 



it follows that 



(xZyOZ(xyZO) 
(xZ 0)Z (xZ yO 



(3) Derive: 

kab • k'ab Z (kab ' k'ab)'; (kab + k'ab)' ^ kab + k'ab I 
kab ■ Wab Z (kab * Wab)'; ((kab + Wab)' Z kab + Wab)'; 
(k'ab • W'ab Z (k'ab * w'ab)0'; (k'ab + w'ab)' Z k'ab + w'ab) 
(kab * w'abZ (kab " w'ab)0'; ((kab + w'ab)' Z kab + w'ab)'- 

(4) Show that 

O^ab iSab Z 0, Qlab Tab Z 0, CtC. 

1 Z a'ab + iS'ab. 1 Z a'ab + Vab. etC. 

and that 

a'ab ^'ab 7'ab e'ab T^ba Z 

1 Z a'ab + /3'ab + 7'ab + e'ab + I'ba 

(5) Assuming Xab=Xab * Xab, Xab = Xab + Xab, 

show that 

a'ab /5'ab =7ab + Cab + Tba, "'ab T^ab =i3ab + €ab + 7ba, CtC. 
«ab + /3ab =7'ab c'ab 7'ba, «ab + 7ab =/3'ab ^'ab 7'ba, etC. 

(6) Derive the general result: 

(ka.bZ Wa.b)'- 

The comma between the terms indicates that the term order is 
not fixed. Thus ka.b stands for either kab or kba. 



8 A PRIMER OF LOGIC 

(7) From the principle, 

(xZzy(yZz)Z(xZyn 
and the postulate (aaa Z a'aa)'. 

derive {a^^L ^aa)', («aaZ 7aa)', («aa^ €aa)'. 

(8) From the principles, 

(xZy)(yZz)Z(xZz), 
(x'Z x)Z (yZ x). 
(xZ xOZ (xZ y), 

and the postulate, a'aaZ aaa» show that all propositions of the 
form XaaZ yaa. cxcept the three cases in the last example, are 
valid implications, Xaa and yaa representing only the unprimed 
letters. 

(9) By the method of the last example, prove that 
(ofaa Z a'aa)' is the Only invalid implication of the form Xaa Z y'aa- 

(10) Derive seven valid implications in each one of the 
forms x'aa Z yaa and x'aa Z yaa and nine invalid implications of 
each one of the same forms. 

(11) From (€aa Z e'aa)' by the method of example 7 derive 

(€aa Z Ofaa)', (€aa Z ^aa)', (caa ^ 7aa)'. 

(12) From e'aa Z €aa by the method of example 8 derive 
thirteen valid implications. 

(13) Show that (e^a Z e'aa)' is the only invalid implication 
of the form Xaa Z y'aa- 

(14) Derive the following implications: 

1 Z ttaa «'aa Z 1 Z tt'aa «aa Z 

1 Z iS^a /3aa Z 1 Z ^\^ ^aa Z 

1 Z 7^a 7aa Z 1 Z 7'aa 7aa Z 

1 Z e'aa €aa Z 1 Z eaa e'aa Z 

(1 Z a'aa)' («aa Z 0)' (1 Z «aa)' (« aa Z 0)' 

(1 Z /3aa )' (^'aa Z 0)' (1 Z 0aa )' (/3'aa Z 0)' 

(1 Z 7aa )' (7 aa Z 0)' (1 Z 7aa )' (Vaa Z 0)' 

(IZ eaa)' (e'aaZ 0)' (1 Z e'aa)' (eaa Z 0)' 

Some of the postulates in the text (p. 2) of the form, 
(k'ab Z Wab)', (kab Z k'ab)', (k'ab Z kab)', may be established by 



A PRIMER OF LOGIC 9 

reducing them to one of the forms, (k'aa^ Waa)', (k'aa^ Waa)' 
(kaa-^ k'aa)', ctc, cases already considered in preceding exercises 

(15) Establish the invalidity of 

a'ab ^ jSab «'ab ^ Tab 7'ab ^ Tba 

/3 ab -^ 7ab /3 ab -^ €ab 7 ab ^ Cab 

a'ab ^ «ab /3'ab -^ i^ab 7'ab ^ 7ab 

ttab -^ « ab Cab ^ € ab C ab -^ Cab 



CHAPTER II 

§4. At this point in our theory it will be necessary 
to introduce certain indefinables, which we shall call the 
distinctions of better and worse, following a suggestion of 
Sir William Hamilton's.* For our immediate purpose 
it will be enough to define better than and worse than de- 
notatively, establishing an order among the four forms by 
a simple enumeration. Better than and worse than are not 
transitive relations. When we wish to express the rules 
for the deduction of the moods symbolically, we shall 
have to invent symbols to represent worse than (/), doubly 
worse than (//), and trebly worse than (///). This necessity 
is avoided in the verbal statement of the principles of de- 
duction by the words "in the same degree" (see rule 1 
below). 

Definitions, — An e-form is worse than an a-, a /3-, or 
a 7-form. 

A 7-form is worse than an a- or a jS-form. 
A jS-form is worse than an a-form. 

Best, a— jS— 7— € Worst. 

§5. Immediate inference is a form of implication 
belonging to one of the types: 

1. Xab Z yab. 2. Xab ^ yba- 

These differences are known as the first and the sec- 
ond figures of immediate inference respectively. 

The part to the left of the implication sign is called 
the antecedent; the part to the right is called the consequent, 

*Lectures on Logic, Appendix, p. 536. 



A PRIMER OF LOGIC 11 

Since X and y may take on any of the forms, a, jS, 
7, e, there will be sixteen propositions of each type, ob- 
tained from the permutations of the letters two at a time 
and by taking each letter once with itself. Each one of 
the sixteen distinct propositions in each one of the two 
figures is called a mood of immediate inference. The 
rules which follow below, applied to the postulates, will 
yield all the true and all the false propositions of each type. 

Valid Moods. 

1. In any valid mood of the first figure make ante- 
cedent and consequent worse in the same degree. 

2. In any valid mood convert simply in any form 
but 7. 

Postulate: aab ^ ciab. Theorems: The other (6) valid 

moods. 
Invalid Moods. 

1. In any invalid mood of the first figure make 
antecedent and consequent worse in the same degree. 

2. In any invalid mood of the first figure interchange 
antecedent and consequent. 

3. In any invalid mood convert simply in any form 
but 7. 

Postulates:"^ 

{ttab^ jSab}'; {aab/7ab}'; { ttab ^ €ab}'; {7ab^7ba}'. 

Theorems: The other (21) invalid moods. 

§6. We may also formulate rules for the detection 
of the invalid moods. These are: 



*The mark (0 over the bracket is intended to indicate that 
the mood is invalid. 



12 A PRIMER OF LOGIC 

1. If the antecedent be worse than the consequent, 
the mood is invaHd. 

2. If the antecedent be better than the consequent, 
the mood is invaHd. 

Definition. — Distributed terms are those modified, 
either impHcitly or explicitly, by the adjective all, i. e. 
the subject of the a-, 7- and e-form, and the predicate of 
the a- and e-form. 

3. If a term be distributed in the consequent but 
undistributed in the antecedent, the mood is invalid. 

§7. A syllogism is a form of implication belonging 
to one of the types: 

1. XbaycbZ Zca=(xyz)x 

2. Xabycb/ Zca=(xy Z)2 

3. Xbaybc-^ Zca=(xyz)3 

4. Xabybc/ Zca=(xyz)4 

These differences are known as the first, second, third, 
and fourth figures of the syllogism respectively. The two 
forms conjoined in the antecedent are called the premises 
and the consequent is called the conclusion. The predi- 
cate of the conclusion is called the major term and points 
out the major premise, which by convention is written 
first, and the subject of the conclusion is called the minor 
term and points out the minor premise. The term common 
to both premises and which does not appear in the con- 
clusion is called the middle term. 

Since x, y and z may have any one of the values, 
a> 0i T> €, there will be sixty-four ways in each one of the 
four figures, called the moods of the syllogism, in which 
X y Z z can be expressed. There will be consequently 



A PRIMER OF LOGIC 



13 



two hundred and fifty-six cases to consider. Twenty- 
nine of these are vaHd implications; the remaining two 
hundred and twenty-seven are invaHd. From the rules 
and postulates below, all the moods, valid and invalid, 
may be deduced. 

Valid Moods. 

1. In any valid mood of the third figure make a like 
major premise and conclusion worse in the same degree. 

2. In any valid mood of the second figure make a 
like minor premise and conclusion worse in the same degree. 

3. In any valid mood convert simply in any form 
but y. 



Postulates: 


Theorems: 


(aaa)x 


(acta) 2. 3. 4 


(/5ai3),.2.3.4 


(777)1 


(a^/3)x.3.3.4 


(7a7)i.3 


(eye). 


(a77)i.2 


(€ae)i.2.3,4 




(a€e)i. 2.3.4 


(eye). 




(yee),,, 






Invalid Moods, 





1. In any invalid mood of the third figure make a 
like major premise and conclusion better in the same degree. 

2. In any invalid mood of the second figure make 
a like minor premise and conclusion better in the same 
degree. 

3. If the premises and conclusion of an invalid mood 
in the fourth figure are all alike, make them all worse in 
the same degree. 

4. If the premises and conclusion of an invalid mood 
are all alike make the conclusion any degree better or any 
degree worse. 



14 A PRIMER OF LOGIC 

5. If the premises and conclusion of an invalid mood 
are all unlike, interchange them in any order. 

6. In any invalid mood convert simply in any form 
but y. 

Postulates:"^ 



(aa/3)'. 


(a^e)', 


(^77)'3 


(777)'. 


(^06)'. 


{aay)\ 


(aT7)'3 


(Pey)\ 


(7^7)'. 


(m)'3 


iaae)\ 


(a 67)', 


(7a7)'a 


{eay)\ 


(67e)', 


(a/37)'. 


i^ay)\ 


(707)'. 
Theorems: 


iepy)\ 




The other (208) i 


nvalid moods. 







§8. As in the case of immediate inference we may 
formulate rules for the detection of the invalid moods of 
the syllogism. These are five in number. 

1. A mood is invalid if the conclusion differ from the 
worse premise. 

2. A mood is invalid if an a- and a 7- premise be 
conjoined in the antecedent and the middle term be un- 
distributed in the major premise. 

3. A mood is invalid if the middle term be undis- 
tributed in both premises. 

4. A mood is invalid if a term which is distributed 
in the conclusion be undistributed in the premise, 

5. A mood is invalid if each premise be in the 
€-form,** 

*The mark (') over the bracket is intended to indicate that 
the mood is invalid. 

**These rules are, of course, not sufficie'it to declare (7 €6)2,4 
and (eye) 1. 2 invalid.,, in cavSe we decLle to §0 regard them. See 
the concluding remarks of chap. III, 



A PRIMER OF LOGIC 15 



EXERCISES 

If Xa.b-^ Ya.b be denoted simply by (xy), (the comma be- 
tween the terms indicating that the term order and so the figure, 
is not determined), the array of sixteen propositions may be 
constructed thu s : 



aa 


Pa 


ya 


ea 


ay 


/37 


77 


ey 


ae 


^€ 


ye 


ee 



the moods vaUd in both figures being underUned twice, the one 
valid only in the first figure being underlined once. Applying 
the first rule to the postulate, we obtain in succession, (33, 77, 
€€, in the first figure; and converting simply in the consequent 
of those valid in the first figure, except 77, we obtain aa, ^(3, ee 
in the second figure. 

(1) From the rules and postulates for the derivation of the 
invalid moods deduce the remaining twenty-one invahd moods. 

The rules for the immediate detection of the invalid moods 
are all necessary if we can point to at least one example which 
falls uniquely under each rule. They are sufficient if they de- 
clare all the invalid moods to be invalid. 

(2) Construct the set of propositions of immediate inference 
and place after each invalid mood the number of a rule which 
declares it to be invalid. 

(3) Make a list of moods which are declared invalid by the 
first rule and by no other rule, and a list of moods which are 
declared invalid by the second rule and by no other rule. 

(4) Find an invalid mood which is declared invalid by the 
third rule and by no other rule and prove that it is the only 
unique illustration of this rule. 

For those who approach the study of the syllogism for the 
first time, it may facilitate manipulation to point out the general 
effect of conversion in the form of certain rules. 

1. Simple conversion in the major premise changes the 
first figure to the second and conversely, the third figure to the 
fourth and conversely. 



16 A PRIMER OF LOGIC 

2. Simple conversion in the minor premies, changes the 
first figure to the third and conversely, the second figure to the 
fourth and conversely. 

3. Simple conversion in the conclusion changes the first 
figure to the fourth and conversely and leaves the second and 
third figure unchanged. 

It must of course not escape the beginner's notice that the 
effect of converting simply in the conclusion is to interchange the 
premises, since the major term then becomes the minor term and 
the minor term becomes the major term. The conjunctive 
relation of logic being commutative, the order of the premises 
is indifferent, but we agree, as a matter of convention, always 
to write the major premise first. 

(5) From (e7€)i, and the third rule under the valid moods 
alone, deduce (e7e)2, (7e€)2 and {yee)^. 

(6) From the rules and postulates deduce the remaining 
valid moods. 

(7) Assuming only the third rule under the valid moods 
and the rule : in any valid mood of the first figure make a like 
major premise and conclusion worse in the same degree, deduce 
all the remaining valid moods from iaaa)i, (777)1 and (0:77)1. 

(8) Assuming only the second and third rules under the 
valid moods deduce the remaining valid moods from {aaa)x, 
(777)1. (70:7)1 and (€7e)i. 

(9) From {e^e)\ alone deduce seventy-eight other invalid 
moods. 

(10) From {a^eYx alone deduce twenty-three other invalid 
moods. 

(11) Deduce the invalid moods in the firs" figure which have 
a 7-minor premise. 

The rules for the immediate detection of the invalid moods 
are sufficient, if they declare all the moods not already found 
to be valid to be invalid. They are all necessary if we can point 
to at least one example which falls uniquely under each rule. 

(12) Construct the array of the syllogism and place after 
each invalid mood the number of a rule that declares it to be 
invalid. 



A PRIMER OF LOGIC If 

(13) Show that it follows from one of the rules alone that 
two ^-premises do not imply a conclusion. 

(14) Prove that there are only two moods which illustrate 
the second rule uniquely. 

(15) Make a list of examples which fall uniquely under 
each one of the rules. 



18 A PRIMER OF LOGIC 



CHAPTER Hi 



§9. In this third chapter it is proposed to completely 
define the relationships of better and worse by deducing all 
the true and all the false propositions into which these 
relationships may enter and then to give a complete ex- 
pression in the language of symbols of the rules for the 
deduction of the moods of immediate inference and the 
syllogism. 

§10. Let us, first of all, invent symbols to denote 
worse than, doubly worse than, and trebly worse than^ i. e. 

X / y =x is worse than y, 

X // y =x is doubly worse than y, 

x///y = x is trebly worse than y, 

and let us add the following: 

Definition. — In the propositions, x / y, x//y, and 
x///y, X is called the inferior, y the superior form. 

Since x and y may take on any of the four forms 
a, /3, 7, e, there will be sixteen possible propositions of each 
type, x / y, X //y and x///y, obtained by permuting the 
letters two at a time and by taking each letter once with 
itself. The following postulates and principles will yield 
all the valid moods of each type. We have assumed four 
principles here because the principles for the deduction of 
the invalid moods may be derived from these four as 
theorems. 

Principles: 

i. (x / y) (y //z) Z (x///z) iii. (x / z) (y///z) Z (y //x) 
ii. (x / y) (z //x) Z (z///y) iv. (x / z) (y // z) Z (y / x) 

Postulates: jS/a; e / y, 7 //a. 

Theorems: 7 / iS; € // /3; e///a. 



A PRIMER OF LOGIC W 

We may also formulate rules for the derivation of 
the moods. It will then be necessary to assume one pos- 
tulate only. 

Definition: — In the propositions, x / y, x // y and 
x///y, the relation connecting x and y is known as the 
worse-relation. 

Definition: — Trebly worse (///) is worse than doubly 
worse (//) and doubly worse than worse (/). Doubly 
worse (//) is worse than worse (/). 

The rules are: 

1. In any valid mood make superior and inferior 
form one degree better or one degree worse. 

2. In any valid mood make inferior form and worse- 
relation one degree better or one degree worse. 

Postulate: Theorems: 

p / a. The other (5) valid moods. 

The invalid moods of each type may be derived from 
the following postulates and principles: 

Principles:"^ 

i. (x / y) (x///z)'Z(y//z) 

(X///Z)' (y//z) Z(x / y) 
ii. (x / y) (z///y)'Z(z//x) 

(z///y)' (z//x) Z(x / y) 
iii. (x / z) (y//x)'Z(y///z) 

(y//x)'(y///z) Z(x / z) 
iv. (x / z) (y / x)'Z(y //z) 

(y / x)' (y//z) Z(x / z) 

*These principles follow from those used for the deduction 
of the valid moods by (xyZ z) Z (xz' Z y'). 



20 A PRIMER OF LOGIC 

Postulates: Theorems: 

(a///e)'; (^///e)'; (t/Z/c)'; The other (36) invalid moods. 

(6///6)'; (6///T)'; (e///0)'. 

As in the case of the valid moods, rules may be form- 
ulated for the derivation of the invalid moods. Here it 
will be necessary to assume only three postulates. The 
rules are: 

1. In any invalid mood make superior and inferior 
form one degree better or one degree worse. 

2. In any invalid mood make inferior form and worse- 
relation one degree better or one degree worse. 

3. In any invalid mood make superior form three 
degrees worse. 

Postulates: Theorems: 

(a / t)'; (7 / a)'; (c / a)'. The other (39) invalid moods. 

§11. Having now completely defined the relationships 
of better and worse by deducing all the propositional forms 
into which these relationships may enter, there remains 
for this chapter only one other task, which is to deduce 
symbolically the moods of immediate inference and the 
syllogism. 

§12. From the postulates and principles, which are 
given below, all the moods, valid and invalid, of immediate 
inference may be deduced. 

Principles: 

i. (y / X) (Xa b / Xa b) ^ (ya b ^ ya b). 

iv. (x Z y) (y Z z) Z (x Z z). 
Postulates : Theorems : 

aab -^ ttbal jSab ^ i^ba; €ab ^ Cba- Thc othcr Valid moods. 



A PRIMER OF LOGIC 21 

Principles :''^ 

ii. (X / y) (Xa b Z Xa b) ^ (Xa b / Ya b)' 
(x//y) (Xa b / Xa b) / (Xa b / Ya b)' 

iii. (y / X) (Xa b Z Xa b) / (Xa b -^ Ya b)' 
(y//x) (Xa b Z Xa b) / (Xa b Z Ya b)' 

V. (xZ y) (xZ z)'Z (yZ z)' 
(xZ z)' (yZ z) Z(xZ y)' 

Postulates : Theorems : 

(ttab ^ €ab)^ (€ab^ ttab)') (Tab ^ 7ba)'. The othcr invalid 

moods. 

§13. All the valid and invalid moods of the syllogism 
may be deduced from the assumptions which follow. 
The right to convert simply in any form but 7 is ex- 
pressed under v and vi. It will be evident that some of 
the postulates might have been saved at the expense of 
introducing new principles, and conversely. The first 
two principles for the deduction of the invalid moods 
under iv are theorems from the ones that have gone be- 
fore under i, by (xy Z z) Z (xz' Z y'). 

Principles: 

i. (y / X) (Xba Zbc / Xca) Z (yba Zbc Z Yea) 
(y / X) (Zab Xcb / Xca) ^ (Zab Ycb ^ Yea) 

V. (xy Z z) (z Z w) Z (xy Z w) 
(xy Z z) (w Z x) Z (wy Z z) 
(xy Z z) (w Z y) Z (xw Z z) 

*These principles, except the second under iii are really 
special cases of principles i and iii under the syllogism, obtained 
from the latter by making b = c, the primed part of the ante- 
cedent in iii becoming unprimed in the special case. 



11 A PRIMER OF LOGIC 

Postulates: (aaa)i; (777)1; (e7e)i- 
Theorems: The other (26) valid moods. 

Principles : 

ii. (y / x) (z / y) (xyZ z)' Z (xz Z y)' 

(y / x) (z //y) (xyZ z)' Z (xz Z y)' 

(y //x) (y / z) (xyZ z)'Z(xzZ y)' 

(y///x) (y / z) (xyZ z)'Z(xzZy)' 

(y / x) (z / y) (xyZz)'Z(zyZx)' 

(y / x) (z//y) (xyZz)'Z(zyZx)' 

(y//x) (y / z) (xy Z z)' Z (zy Z x)' 

(y///x) (y / z) (xyZ z)'Z (zyZ x)'. 

iii. (X / y) (Xa,b Xb.c ^ Xca)' Z (Xa.b Xb.c Z yea)' 
(X // y) (Xa.b Xb.c Z Xca)' Z (Xa .b Xb.c Z yea)' 
(y / X) (Xa.b Xb.c ^ Xca)' Z (Xa.b Xb.e Z yea)' 

iv. (X / Z) (yab Xeb ^ Xca)' ^ (yab Zeb ^ Zca)' 
(X / Z) (Xba ybc ^ Xca)' ^ (Zba ybc Z Zea)' 
(y / X) (XabXbc/ Xca)'Z (yab ybc Z yea)' 

vi.* (xy Z z)' (w Z z) Z (xy Z w)' 
(xy Z z)' (x Z w) Z (wy Z z)' 
(xy Z z)' (y Z w) Z (xw Z z)' 

(aa^)'x (a/37)'x (ae7)'x (^77)'3 (t^t)'^ (ea7)'x (e77)'3 
(aa7)'x (a^e)', (^a7)'x {^ey)\ (777)'. (e^7)'x (676)^3 
(aae)'. (a77)'3 (/3/36)'x (7a7)'a {ley)'. {e^e)\ {eea)\ 

Theorems: 
The other (206) invaUd moods. 

*Principles v and vi are of course not independent. The 
first under v is a variation of transitivity, the third a variation 
of the second by xyZ yx. Those under vi follow Crom those 
under v by (xyZ z) Z (xz'Z y'). Principles v under immediate 
inference follow from transitivity by the same principle. 



A PRIMER OF LOGIC 23 

§14. We have already pointed out, (note p. 2), that 
the product 7a. b ^a.b does not vanish in general if we allow 
the possibility of the limiting values and 1 for the terms. 
Under these conditions, (7 €6)2,4 and (€7e)i,2 are not valid 
moods of the syllogism, for they become 7oi eo.iZ 0, for 
a = c = and b = 1. A logic, which recognizes these limiting 
values of the terms, will have to postulate {ey e)\, say, 
which yields, (e7€)'2 and {yei)'^,^. 

The only change, which we should then have to make 
in chapters II and III, would be to replace (777)^2 among 
the postulates by {eye}\, from which (777)'2 follows as a 
theorem, and to subtract (e7e)i,2 and (7 €6)2,4 from the list 
of valid moods. 

This logic, which might be called non-Aristotelian, or 
semi- Aristotelian, or imaginary logic, is more general than 
the ordinary or classical logic and includes the latter as 
a special case, becoming, in fact, identical with it when the 
field of its application is narrowed so as to exclude "nothing" 
and "universe" as limiting values of the terms. One 
principle, which is true in the special, but not in the general 
case, is: 

(y / X) (Xba Zcb Z Xca) / (yba Zcb / yea), 

and this principle may be regarded as the differentiating 
character of the two cases. If we had chosen to assume 
it, instead of the first principle under i, we could have saved 
the third postulate, but the second principle under iv 
would not then have followed as a theorem. 

The definitions of chapter I, §3, hold for both cases; 
the only change to be made in implications V in order to 
make them true in the more general logic, will be to replace 
7aa = by 7aa4= 0, and this property has not been made use 
of in defining the 7-form. 



U A PklMER OF LOGIC 

The Aristotelian forms, A, E, I, O, (see Appendix I), 
will yield only eight valid moods of the syllogism, under the 
new condition, instead of the tv/enty-four valid moods 
commonly recognized. They satisfy all the conditions of 
maximum simplicity in the special or classical instance — 
they are the best possible forms to choose for the con- 
struction of an Aristotelian logic — but they fail in the gen- 
eral instance, for they then lose their peculiar advantage, 
that, corresponding to any member of the set there should 
be another member of the set which represents its contra- 
dictory. 



A PRIMER OF LOGIC 25 



EXERCISES 

By the aid of the principles, 

(xy Z z) (z Z w) Z (xy Z w) 
(xy Z z) (w Z x) Z (wy Z z) 
(xy Z z) (w Z y) Z (xw Z z) 

we are enabled to convert in either premise or the conclusion. 
The example which follows will illustrate the method. 

(Tba «cb ^ Tea) («cb ^ «bc) Z (7ba «bc Z 7ca) 

(1) From (e7e)i derive (t€ 6)2,4. 

(2) From the principles and the postulates in the text de- 
duce the remaining valid moods. 

(3) From the postulates, {aaa)i, (777)1, (0:77)1 and the prin- 
ciple (y/x) (xba Zcb Z Xca) Z (yba Zcb ^ Yea) dcducc the remaining 
valid moods. 

If we identify the subject and predicate of the conclusion 
in the mood, {(3a^)^, we obtain /3ba«baZ 0, (chapter I, impli- 
cations v). By the aid of (xyZ 0) Z (xZ y') it follows that 

Oi&h^ i3 ab. 

(4) Deduce as many as possible of the propositions of the 
form, XabZ y'ab, (chapter I, p. 2) by identifying subject and 
predicate in the conclusion of the valid moods of the syllogism. 

The principles, 

(xyZ z)'(wZ z)Z (wyZ w)' 
(xyZ z/ (xZ w)Z (wyZ z)' 
(xyZ z)'(yZ w) Z (xw Z z)' 

enable us to convert in either premise or the conclusion. 

(5) From {^^^Yi, derive the invalidity of this mood in the 
other figures. 

(6) From {e^e)\ alone and principles iii, iv and vi deduce 
seventy other invalid moods. 

(7) From {a^yYi alone and principles ii deduce nineteen 
other invalid moods. 



26 A PRIMER OF LOGIC 

(8) Deduce the invalid moods in the third figure, whose 
conchision is in the 7-form. 

(9) Deduce the invahd moods in the fourth figure, whose 
major premise is in the 7-form. 

(10) From 7771 alone, deduce forty-six valid implications 
of the form La.b Mb.c ^ N'ca, — L, M and N representing only 
the unprimed letters. 

(11) Assuming €7 €1.2 and 7 €€2,4 to be invalid, show that 
777 1 yields only thirteen valid implications of the form given 
in the last exercise. 

(12) Assuming (^7. my'V . (Me)\ (77^0'^ (77^0^ (7770'x 
(77 0'. (e7/30\(€7€0'3 (^^/^O.'x {eee')[. 

deduce sixty-nine other non-implications of the same form. 

Any non-implication of the form. Lb, a Mc.b ^ N'ca, which 
contains an a-form may be proven invalid by identifying terms 
in the a-form. Thus jSbajSbc^ «'ca reduces to jSba^ jS'ba for 
c = a; 7ba«cb^ 7'ca reduces to 7baZ 7'ba for c = b, etc. 

(13) Establish the invalidity of the thirty-four non-impli- 
cations of the form Lb. a Ma.b^ N'ca not accounted for in the 
preceding exercise. 

(14) Show that there are thirty-six, and only thirty-six, 
distinct valid implications of the form La,b Mb.c Nc.a-^ 0, — • 
L, M and N representing only the unprimed letters, a, /8, 7, e. 

(15) Derive indirectly the (7770^1 of exercise 12. 

A certain number of the postulates for the derivation of the 
invalid moods of the syllogism (p. 21) may be shown indirectly 
to be invalid by reducing them to invalid moods of immediate 
inference. Thus (aa(S)j_ reduces to aab -^ oi'ab when the terms 
in the conclusion are identified, and {a^y)i reduces to j8ca -^ 7ca 
when we identify terms in the major premise and suppress the 
part Qiaa (see chap. IV). 

(16) Establish the invalidity of 



(aay)^ 


(a;«e)i 


We). 


(«^6). 


(a €7) I 


Way), 


(€0:7)1 


(^77)3 


(70:7)2 



A PRIMER OF LOGIC 27 

Most of the other postulates for the deduction of the invaHd 
moods of the syllogism may be reduced by the method of the 
following example: 

Suppose that {€(3y)j. is valid. Now (e/37')i is valid by a 
preceding exercise. 

. ' . ( Cba /5cb / Tea) ( €ba ^cb ^ 7'ca) ^ (tba /?cb ^ 0) 

since (xZ y) (xZ yO Z (xZ 0). 
Consequently €ba i^cb ^ Cca- 
If now we postulate (e^eYi, it follows that {e^y)\. 

(17) Establish the invalidity of 

(/3t7)3 (7/37)2 (777)4 

(767)2 (^77)3 (e€a)i 

The postulates {^ei)\ {e^f)\ and (€7e)'3 that remain (p. 21) 
may be reduced by the following method: 

(jSba |8cb ^ e'ca)' (jScb ^ 7'cb) ^ (^ba 7'cb ^ c'ca)' 

by a principle under vi and the postulate of a preceding exercise 

(j8ba 7'cb ^ e'ca)'^ (/3ba 6ca ^ 7cb)' 

which yields {^ey)\ by simple conversion in the major premise. 

(18) Establish the invalidity of ( e^S e)i and {eye)^ 

Any non-implication of the form La.b Mb.c ^ Nca, which 
contains an cf-premise, may be reduced to an invalid mood of 
immediate inference, and so shown to be invalid, by identifying 
terms in the a-premise. All of the other invalid moods may be 
derived from the postulates of exercise (12), the forms of im- 
mediate implication given in chapter I, the principles iv and vi 
of chapter III, together with (xy Z z) Z (xz'Z y') and fxy Z z) Z 
(z'yZ x'). 

19. From the postulates of exercise (12) deduce all the 
non-impHcations of the form La.b Mb.c-^ Nca, without making 
use of principles ii and iii of this chapter. 

20. Show that there exist no valid implications of the 
form L'a.b Mb.c Z Nca or La.b M'b.c Z Nca and consequently 
none of the form L'a.b M'b.c Z Nca or L'a.b M'b-c Z W^^, 



CHAPTER IV 

§15. The sorites is a form of implication of the 
general type:* 

Xi(l,2) X2(2,3) X3(3,4)— Xn-i(rW, II) Z Xn(n l), 

in which the number of terms is greater than three. 

Certain valid moods of the sorites can be constructed 
from chains of valid syllogisms. Thus the chain of syl- 
logisms: 

a{i,2) a(2,3) Z a(3i), 
a(3i) a(3,4) Z a(4i), 
a(4i) a(4.5) Z a(5i), 

will yield a valid sorites, viz : 

a(i,2) a(2.3) a(3,4) a(4.5) Z a(5i), for 

{ a(i,2) a(2,3) Z a(3i) } Z { a(i.2) a(2,3) 0(3.4) Z a(3i) 0(3.4) } 

.*. a(i,2) 0(2.3) 0(3.4) Z 0(41), by the second syllogism and 

the principle of transitivity. 

{ 0(1,2) 0(2,3) 0(3,4) Z 0(41) } Z 

{ 0(1,2) 0(2,3) 0(3.4) 0(4,5) Z 0(41) 0(4,5) } 

.*. 0(1,2) 0(2,3) 0(3,4) 0(4.5) Z 0(51), as before. 

Consequently in general, if 

Xi(l.2) X,.(2.3) Z X3(3l) 
X3(3l) X4(3.4) Z X5(4l) 
Xs(4l) X6(4.5) Z X7(5l) 



X2n.5(n-i 1) X2ii.4(n-i, n) Z X2n-3(ni) 



*In this chapter it will be more convenient to employ the 
notation x(ab) for Xab or x(i.2) for Xi.2. The comma between 
the terms means that the term order is not settled. 



A PRIMER OF LOGIC 29 

be a chain of valid syllogisms, then 

Xi(i,2) X2(2,3) x^(3a)— X2n-4(n- 1 , n) Z X2n-3(ni) 

is a valid mood of the sorites. It remains to be proven 
that the only valid moods that exist can be constructed 
from chains of valid syllogisms. The proof depends on 
the following principles. 

Principle i. — A valid mood of the sorites, which has 
one premise of the same form as the conclusion, will re- 
main valid, when as many of the other premises as we 
desire are put in the a-form. 

Principle ii. — A valid mood of the sorites will remain 
valid, when as many terms have been identified as we desire. 

Principle iii. — An a-premise, whose subject and predi- 
cate are identical, may be suppressed as a unit multiplier. 

Principle iv. — A valid mood of the sorites, none of 
whose premises has the same form as the conclusion, will 
remain valid, when as many premises as we desire are put 
in the a-form. 

Theorem i. — There exists no valid mood of the sorites, 
in which none of the premises has the same form as the 
conclusion. 

For (principle iv) put all the premises after the first 
in the a-form. Then by identifying terms (principle ii) 
the mood of the sorites can be reduced (principle iii) to an 
invalid syllogism of the form: 

Xi(l.2) a(2.3) Z Xn(3l). 

Conclusion in the a-form. 

At least one of the premises is in the a-form (theorem i). 
If one of the remaining premises, Xr(s - i, s), be not in the 
a-form, put each one of the other premises in the a-form, 



30 A PRIMER OF LOGIC 

if all but Xr be not already in that form (principle i). 
Then by identifying terms (principle ii) the mood of the 
sorites will reduce (principle iii) to an invalid syllogism 
of the form: 

Xr(s- 1, s) a(s, s + i) Z a(s + i s - i), 
or a(s - 2, s - i) Xr(s - i, s) Z a(s s - 2). 

Consequently all the premises are in the a-form if the 
mood of the sorites is valid and the sorites is of the general 

type: 

a(i.2) 0(2.3)— a(n- 1, n) Z a(ni), 

which can be constructed from the chain of valid syllo- 
gisms : 

a(2.i) 0(3.2) Z a(3i), 

a(3i) 0(4,3) Z 0(41), 

0(41) 0(5.4) Z 0(51), 



o(n - 1 1) o(n, n - 1) Z o(n 1). 

Conclusion in the P-form, 

At least one premise, Xt, is in the /S-form (theorem i), 
and all the other premises are in the o-form. For suppose 
one of the other premises Xr (s - 1, s) were not in the o-form. 
Put all the premises (principle i) except Xt and Xr in the 
a-form. Then by identifying terms (principle ii) the mood 
of the sorites will be reducible to an invalid syllogism 
(principle iii) of the form: 

i8(sTT, 7^) Xr(s, i^) Z |S(S ^T^), 
or Xr(s, s- 1) i3(s, sTi) Z iS(sT~i s- 1). 

Consequently the sorites must be of the form: 

0(1,2) 0(2,3) — o(s, s - 1) i3(s + i, s) o(s + i, s+2) — o(n- 1, n) 



A PRIMER OF LOGIC 31 

Z /3(n i), which can be constructed from the chain of 
syllogisms : 

a(i,2) 0(2,3) Z a(3i) 
a(3i) 0(3,4) Z a(4i) 



a(s - 1 1) a(s - 1, s) Z a(s 1) 
a(si)/3(s, ^)Z p(^i 1) 
iS(^ 1) a(iTi, sT^ ) Z j8(m^ 1) 



/3(n-i 1) a(n-i, n) Z iS(n 1). 

Conclusion in the y-form. 

At least one of the premises is in the 7-form (theorem i). 
Each 7-form in the antecedent must present its terms in 
the order (s s - 1). For suppose that 7(3 - 1 s) should 
appear as one of the premises. Put each one of the 
remaining premises in the a-form (principle i). Then by 
identifying terms (principle ii) the sorites will reduce to 
an invalid syllogism (principle iii) of the form: 

7(s - 1 s) a(s, s + 1) Z 7(5 +1 s - 1), 
or a(s - 1, s - 2) 7(3 - 1 s) Z 7(3 s - 2). 

Pursuing the same reasoning as before it can be shown 
that no /3- or e- premises can occur. One form of this 
sorites may consequently be 7(21)— 7(11 n- 1) Z 7(ni), which 
can, in fact, be constructed from the chain of valid syl- 
logisms : 

7(21) 7(32) Z 7(31) 

7(31) 7(43) Z 7(41) 



7(n-i 1) 7(n n- 1) Z 7(n 1). 

All the other forms of valid sorites with a 7-conclusion 
are obtained from the above type by transforming one or 



32 A PRIMER OF LOGIC 

more of the premises into the a-form in every possible 
way under the restrictions of theorem i. Each one of 
these types can be built up from a chain of vaHd syllo- 
gisms each member of which has one of the forms: 

(777)1, (a77)i.2, or (7a7)x.3. 

Conclusion in the e-form. 

At least one of the premises is in the e-form (theorem i), 
and there is not more than one e-premise. For, if there 
are two or more e-premises, put all the premises but two 
of the e-premises in the a-form (principle i). Then by 
identifying terms (principle ii) we Avill come upon an invalid 
syllogism (principle iii) of the form: 

€(s-i, s) €(s, s+i)Z e(s + i s - 1). 

There can be present no /3-premise. For suppose 
Xr (s, s - 1) to be a /3-premise. Put all the premises except 
Xr and the e-premise in the a-form (principle i). By identi- 
fying terms (principle ii) we will come upon an invalid 
syllogism (principle iii) of the form: 

i8(s, s - 1) e(s, s+i) Ze(s + i s - 1), 
or e(s -1,5-2) i8(s, s - 1) Z e (s s - 2). 

Any 7-premise coming after the e-premise must pre- 
sent its terms in the order (s s - 1). For suppose 7(3, s - 1) 
coming after the e-premise to present the term order 
(s - 1 s). Put all the premises except 7(3 - 1 s) and the 
e-premise in the a-form (principle i). Then by identifying 
terms (principle ii) we v/ill come upon an invalid syllogism 
(principle iii) of the form: 

e(s - 1, s - 2) 7(5 - 1 s) Z e(s s - 2). 
Any 7-premise coming before the e-premise must 
present its terms in the order (s - 1 s). For suppose 7(3, s - 1) 



A PRIMER OF LOGIC 33 

coming before the e-premise to present the term order 
(s s - i). Put all the premises except 7(3 s - i) and the 
e-premise in the a-form (principle i). Then by identifying 
terms (principle ii) we will come upon an invalid syllogism 
(principle iii) of the form : 

7(3 s - 1) e(s, s + i) Z €(s + i 3 - 1). 

One form of this sorites may be, consequently, 

7(12) 7(23)— 7(3 - 2 s - 1) e(3, 3 - 1) 7(3+1 s) — 7(n n- 1) Z 
€(n 1), which can, in fact, be constructed from the chain 
of valid syllogisms: 

7(12) 7(23) Z 7(13) 
7(13) 7(34) Z 7(14) 



7(1 3 - 2) 7(s - 2 3 - 1) Z 7(1 s - 1) 

7(1 3 - 1) €(3, 3 - 1) Z £(3 1) 

€(s 1) 7(3 + 1 3) Z e(s+i 1) 

€(n - 1 1) 7(n n - 1) Z e(n 1) 

All the other forms of valid sorites with an e-conclusion 
are obtained from the above type by replacing one or more 
of the 7-forms by a-forms in every possible way. Each 
new type can be constructed from a chain of valid syllo- 
gisms, each member of which has one of the forms: 

(a€e)i.2,3,4, (€ae)i.2,3.4, (7e€)2,4, (e7e)i,2. 

There exist, consequently, no valid moods of the sorites 
which can not be constructed from chains of valid syllo- 
gisms.* 

*If (7 €6)2,4 '^i^d (€7e)i,2 are to be regarded as invalid moods, 
(see the concluding remarks of chapter III), then it can be shown 
at once that no 7-premise can occur when the conclusion is in 
the €-form. The general form of such a sorites will be. 



M A PRIMER OP LOGIC 



EXERCISES 

(1) Construct a valid sorites from the chain of valid syllo- 
gisms: 

«2i732^ 731. 

731 «43^ 74i» 

741 7s4^ 7si' 

(2) By the aid of the principles of chapter IV, reduce the 
valid sorites, aai 732 «43 7s4^ 7si. successively to each one of the 
three valid syllogisms of example 1. 

(3) Prove the invalidity of the sorites, 

721 y^i 734 Ts4 ^ 7si- 

(4) From what chain of valid syllogisms can the sorites, 
ai,2 723 63,4 7s4 ae.s -^ C6i be constructed? 

(5) If (e7€)i.2 and (7 €6)2,4 be regarded as invalid moods of 
the syllogism, (see the concluding remarks of chap. Ill), prove 
the invalidity of the sorites, 

7 12 733—73-2 8-1 e s, s-i 7s-|-l s~7n n-i ^ €ni 



a(i,2) a(2,3) — a(s - 1. s)€ (s, s + i) a (s + i, s+2) — a{n- 1, n)Z €(ni) 
which can be built up from the chain of syllogisms, 

a (1, 2)01(2 ,3) Z a(3i) 

Q!(3i)a(3,4) Z q:(4i) 



a(s - 1 i)q:(s - 1 , s)Z a{s\) 
a(si)e(s, s+i) Z €(s+i 1) 
€(s+i 1) a (s-f 1, s+2)Z €(s+2 1) 



€(n - 1 1 )a (n - 1 , n) Z € (n 1) 



APPENDIX I 

On the Simplification of Categorical Expression and the 
Reduction of the Syllogistic Figures 

If a and b represent classes, there are four ways in 
which they may be related categorically, the one standing 
for subject, the other for predicate. These four forms of 
relationship are always represented by the letters, A, 
E, I, O, i. e. 

Aab=all a is b, 
Eab = no a is b, 
lab =some a is b, 
Oab =not all a is b. 

Historical efforts have been made to reduce the num- 
ber of these relationships. If symbols be invented to 
denote some a (a) and not-a (ai), the last three may be 
represented by means of the first, for: 

Eab=Aabi> Iab=Aab> Oab=A&bi» 

But an essential difference is here left undistinguished 
and the number of necessary forms will not have been 
reduced by this device. If a new symbol be employed for 
all a (a) and another for the copula ^'s ( Z ), we shall have: 

Aab=aZ b, 
Eab=aZ bi, 
lab =^Z b, 

Oab=^Z bx. 

The four separate categorical forms have, accordingly, 
been gotten rid of at the cost of introducing four new unde- 
fined symbols, so that no economy of our indefinables has 
been effected. 



36 A PRIMER OF LOGIC 

It is to be observed that the word some, which is im- 
pHcit or explicit in the meaning of part of each proposition, 
means some at least, possibly all. Another set of proposi- 
tions, in which some is to mean some at least, not all, may 
be used to replace the traditional ones. These other forms 
are: 

aab=all a is all b, 

/3ab=some a is some b, 

Tab = all a is some b, 

€ab=no a is b. 

In addition we shall have to employ 
the hypothetical form, 

Xab / yab =Xab impHcS yab, 

{ Xab / yab}' = Xab docs not imply yab, 
the conjunctive form, 
Xab * yab = Xab and yab, 
the disjunctive form, 

Xab + 3'ab=Xab Or yab- 

Each member of the set. A, E, I, O, may be expressed 
in the members of the set, a, /3, 7, e, and conversely, so 
that the two are, in fact, logically equivalent, although 
each one has certain advantages peculiar to itself. 

The members of the second set have this property, 
that, if one is true, then all the others are false.* We 
assume, accordingly, the 

Postulates: a^h ^ jS'ab i^ab ^ 7'ab Tab ^ T'ba 
ttab ^ T'ab ^ab ^ c'ab 
ttab -^ e ab Tab ^ € ab 

*Provided we exclude the limiting values and 1 for a and b. 
The ordinary definition.s of these limits allow 701 and €01 be true 
together. 



A PRIMER OF LOGIC 37 

from which follow, by the principle of the denial of the 
consequent, the 

Theorems: eab ^ a'ab Tab ^ a'ab 

€ab ^ jS'ab Tab / jS'ab 

€ab -^ T ab P&h ^ 0. ah 

Consequently* 

aab'iSab=0 i3ab*Tab=0 Tab'Tba=0 

ttab *Tab=0 ^ab * eab=0 

Ctab * €ab=0 Tab * €ab=0 I 

From the Definitions:*'^ 

Aab =aab + Tab 

Eab = €ab 

lab = Ctab + i^ab + Tab + Tba II 

Oab = €ab + /?ab + Tba 

we obtain immediately*** 

Gab =Aab ' Aba 
/5ab = lab * Oab * Oba 
Tab =Aab * Oba 
^ab ^^ -t!>ab 

It is an advantage of the forms of the original set, 
an advantage which the set, a, /3, y, e, does not possess, 
that the contradictory of any letter is represented by a 
single other letter of the set. Suppose that we were to 

*q: i8 = reads: a[is true) and ^{is true) is impossible. 

**A, E, I, O are simply the sums given in equations II. 
That they are the traditional Aristotelian forms, is only an 
accident of the reader's application. Hence equations II are 
definitions and not postulates. 

***Multiplying out the sums in II as if they were ordinary 
polynomials, applying the results of I, and assuming that a, ^ 
and e are simply convertible. 



38 



A PRIMER OF LOGIC 



combine this advantage with that of simple convertibility 
in a new set of forms. 

To do this it would seem to be enough to subtract 
from the meaning of Aab the part Tab, (equations II), and 
to add this part to the meaning of Oab.* Our new set of 
forms becomes: 

Ctiab =aab 

Ciab ^^ ^ab 

tiab =aab + jSab + Tab + Tba HI 

Oiab = Cab + /3ab + Tab + Tba 

the analogues of the old letters being represented by the 
corresponding Greek vowels. 

From equations I and III, and remembering that the 
sum of ttab, i^ab, Tab» ^ab, Tba makcs up the propositional 
"universe," the results of the following table, yielding all 
the moods of immediate inference, will easily be seen to 
hold. 



True 


Implies the 

truth of 

only 


Implies the 

falsity of 

only 


False 


Implies the 

truth of 

only 


Implies the 

falsity of 

only 


a 


a, t 


e, 


a 





a 


e 


e, 


a, t 


e 


t 


€ 


L 


I 


€ 


I 


e, 


a, (, 








a 





a, I 


€, 



An induction of these results shows that a = o', 
<? = a', € = t', t = e', and that, consequently, contradictory 

*Here would seem to be another instance of the manner in 
which the language of symbols may free a science from the 
accidents imposed upon its development by the language of 
speech. The last two members of the new set have apparently 
no simple verbal expression. 



A PRIMER OF LOGIC 



39 



pairs are a, o and e, t. Likewise contraries are a, e; 
suhcontraries are t, o; subalterns are a, t and e, o. 

If we define an affirmative form as one that becomes 
unity when subject and predicate have been identified 
and a negative form as one that becomes unity when sub- 
ject and predicate have been made contradictory, then it 
is a result of the following 

Postulates:^ i 



IS a true proposition, 



Theorems: 



and equations III, that a and t are affirmative and that e 
and are negative forms. 

If a distributed term be one modified by the quanti- 
tative adjective all, it will be seen that a and € distribute 
both subject and predicate, while t and o distribute neither. 
These results are summarized in the table below, the 
distributed terms being underlined. 





Affirmative 


Negative 


Universal 


ttab 


€ab 


Particular 


tab 


^ab 



*x is true is to be represented by x = l, x is false by x = 0. 
(See Boole, Investigation of the Laws of Thought, ch. XI, p. 
169). The theorems follow by equations I, and equations III 
become as a result of them, aaa=L Caa = 0, taa = l, <?aa = 0, 
aaa = 0, €aa=l, iaa = 0, Oafi=l. Employing the usual notation, 
a = not-a. 



40 A PRIMER OF LOGIC 

The traditional rules for detecting the invalid moods 
of the old syllogism, constructed from the set A, E, I, O, 
hold for the new syllogism, built up out of the forms, 
a, e, I, 0. These rules are: 

1. Two negative premises do not imply a conclusion. 

Ex. €ba €cb ^ €ca- 

2. Two affirmative premises do not imply a nega- 
tive conclusion. Ex. aba acb ^ Cca- 

3. An affirmative and a negative premise do not 
imply an affirmative conclusion. Ex. aba Ccb ^ ctca- 

4. Two premises, in neither of which the middle 
term is distributed, do not imply a conclusion. 

Ex. tba tcb ^ tea. 

5. Two premises, in which a given term occurs un- 
distributed, do not imply a conclusion, in which that same 
term is distributed. Ex. aba tcb ^ ctca. 

The valid moods which remain, and which of course 
are valid in all four figures, since each one of the forms 
is simply convertible, are twelve in number, viz: 

aaa aai aee aeo 

aiL aoo eae eao 

Lai oao €10 Leo 

It has been previously observed, (note p. 2), that 
equations I hold generally only when the limits and 1 
are excluded as possible values of a and b. If these pos- 
sibilities he included, we shall have to assume: 

{Tab -^ e'ab }' and \\ { eab / y'X Y, since 7oi Coi=i= 0. 
Equations I then become: 

aab ' i3ab=0 iSab ' Tab = 

Clab *Tab=0 jSab * €ab = IV 

dab * Cab=0 Tab * Cab 4=0 



A PRIMER OF LOGIC 41 

Under these conditions, the fact which the old logic 
always took for granted, that Eab is the contradictory of 
lab, and Aab the contradictory of Oab, no longer holds 
true. For, while the sum of each of these two pairs of 
forms is the propositional universe, their product is not 
the propositional null, (equations II, IV). In order that 
Eab * lab and Aab * Oab shall vanish for all values of the 
terms, it will be necessary to exclude 7ab Cab from the 
product. A new set of forms, in which part of the meaning 
of tab is subtracted from tab and added to €ab, will satisfy 
this requirement. Let this new set be: 

Ctaab =Ctab 

Caab = Cab + Tab + Tba 

t'2ab =aab + 0ab 

02ab= €ab + jSab + Tab +Tba V 

If a, e, t, 0, be replaced by aa, €2, t2, ^2, respectively 
in the table, (p. 38), all the results of such a new tabulation 
will be seen to hold, (equations IV, V). The same defini- 
tions as given before will make az and ta affirmative, 62 and 
O2 negative forms, (equations V, and the postulates and 
theorems, p. 39), but since 62 distributes neither subject 
nor predicate, €2 t2 O2 and t2 €2 O2 will not be found among 
the valid moods of the syllogism, (see p. 40). The same 
rules (p. 40) for the detection of the invalid moods will 
hold for the new syllogism, but rule 1 is now redundant, 
being a corollary of rule 4. 

It might perhaps appear that our original symmetry, 
(that of equations I), which was interrupted by the neces- 
sity of allowing Toi to stand as a true proposition, could 
be saved by assuming that the null class exhausts no part 
of the universe, i. e. all of nothing is some of every things 
might be regarded as a false proposition. Now Aoa = l, 



42 A PRIMER OF LOGIC 

or aoa + 7oa = l, is Schroder's definition of the null class, 
and Aoa will be a true proposition for all values of a if 
7oi be true, whereas, 7oi = involves Aoi=0. These con- 
sequences lead us to the alternatives of either giving up 
our symmetry, (in equations I), or else of regarding the 
null class as not essential to our algebra. 

It is finally to be noted — what was obvious in the 
beginning — that, while the members of the set, a, e, i, o, 
can be expressed in the members of the set, a, jS, 7, e, 
the latter can not be expressed in the former. Conse- 
quently, an essential difference has been lost, and the 
existence of a completed logic of the new forms would not 
put aside the necessity of working out the logic of the old. 

The attempts of the logician to discover a set of 
categorical forms, which establish a complete symmetry 
among the moods and figures of the syllogism, are as old 
as the science itself. The end would be attained if a new 
set could be selected so as to satisfy the following con- 
ditions: 

1. Each form of the set must be simply convertible. 

2. Corresponding to any member of the set, there 
must occur another which represents its contradictory. 

3. The new set must yield at least one valid mood 
of the syllogism. 

4. Each member of the new set must be represent- 
able in the members of a set already proved necessary 
and sufficient to express all differences, (the set A, E, I, 
O, say), and conversely. 

If Xab be any categorical form, the simplest functions 
of X, which are themselves categorical and which are in 
general simply convertible, are Xab * Xba and Xab + Xba. It 
will be enough, therefore, in order to satisfy condition 1, 



A PRIMER OF LOGIC 43 

to assume as a new set of forms either such a sum or 
such a product of each one of the old forms (A, E, I, O, 
say). 

The equation, {xab * Xba }' =x'ab + x'ba, suggests at 
once what our manner of satisfying condition 2 must be, 
for since the product, Xab ' Xba is the contradictory of the 
sum x'ab + x'ba and x the contradictory of x', if Xab + Xba 
[respectively Xab * Xba] be chosen as one of our new forms, 
x'ab ' x'ba [respectively x'ab + x'ba] must be chosen as one 
of the others. 

Remembering that 

Urab^^-t^ab J^ba^^-t^ab "r li'baj 
and lab =Iab * Iba =Iab + Ibaj 

and that Eab = I'ab and lab =E'ab, 

it follows that our choice of a new set of forms is limited 
to the following two: 

Ai=Aab'Aba> Ei=Eab» (ij 

0i=0ab + 0ba, Ii =Iab, 

A3=Aab + Aba, E.=Eab, (2) 

02=0ab ' Oba, I2 = lab- 

It will be found, however, that the set (2) yields no 
valid moods of the syllogism. Consequently, applying 
condition 3, our choice is seen to be unambiguously re- 
stricted to set (1), which yields twelve moods, valid each 
one in each of the four figures. These will be found to 
be, (dropping the subscripts): 



AAA, 


AAI, 


AEE, 


AEO, 


All, 


A 0, 


EAE, 


EAO, 


I A I, 


OAO, 


EIO, 


lEO. 



It will be impossible, however, to satisfy condition 4, 
since every expression involving the new forms will be 



44 A PRIMER OF LOGIC 

simply convertible. Consequently, an essential difference 
has been left undistinguished, and it will not be possible 
to substitute the new forms for the old. The new forms 
are, in fact, identical with ai, ej, ti, t?i, considered above. 
From this latter discussion and from the discussion 
that has gone before, we conclude, that, if it be necessary 
to retain in our set of forms at least one that is not simply 
convertible, it will be impossible to satisfy the condition 
2 above, unless the null-and one-class be excluded or de- 
fined in some way other than ordinary. 



APPENDIX II 

Historical Note on De Morgan's New Propositional Forms 

In introducing the notion of contradictory terms into 
logic De Morgan discovered two new propositional forms, 
which cannot be directly expressed by means of the A, E, 
I, O relations of traditional logic* Suppose that we de- 
note these two forms by U and V, i. e. 

U (ab) =A11 not a is b, 

V (ab) =Some not a is not b, 

A (ab) =A11 a is b, 

E (ab) = No a is b, 

I (ab) =Some a is b, 

O (ab) =Some a is not b. 

U and V are simply convertible, for (if a = not-a) 
U(ab) =A (ab) =A (ba) =U (ba), (converting in A by the 
principle of contraposition), and 

V (ab) = I (ab) = I (ba) = V (ba), (converting simply in I). 

V distributes both subject and predicate while U 
distributes neither, for V(ab) =0(ab) =0(ba) (converting 
in O by contraposition) and, since O distributes its 
predicate, both a and b are distributed terms; similarly 
U(ab) =A(ab) =A(ba) (converting in A by contraposition) 
and, since A does not distribute its predicate, a and 
b are undistributed terms. 

If an affirmative form be defined as one that becomes 
the subcontrary of itself when the subject and predicate 

*Formal Logic, p. 61. 



46 



A PRIMER OF LOGIC 



have been identified, and a negative form as one that 
becomes the contrary of itself under the same conditions, 
it will be seen that A, I and V are affirmative, that E, O 
and U are negative forms. 

These results are summarized in the following table, 
the distributed terms being underlined. 





Affirmative 


Negative 


Universal 


A(ab) 


E(ab) 


Particular 


I(ab) 


0(ab) 


Indefinite 


V(ab) 


U(ab) 



Below are tabulated all the forms of immediate im- 
plication which hold among the six propositions A, E, I, 
O, U, V. The (144) implications and non-implications 
necessary to establish unambiguously the results of the 
table can be derived from a certain number of postulates 
and the commonly assumed principles of traditional logic. 





Implies 


Implies 




Implies 


Implies 


True 


falsity of 


truth of 


False 


falsity of 


truth of 


A 


E, 0, U. 


A, I, V. 


A 


A. 


0. 


E 


A, I. 


E, 0. 


E 


E. 


I. 


I 


E. 


L 


I 


A, I. 


E, 0. 





A. 


0. 





E, 0, U. 


A, I, V. 


U 


A,V. 


0, U. 


U 


U. 


V. 


V 


U. 


V. 


V 


A,V. 


0, U. 



An induction of these results will show that: 
Contradictory pairs are: A, O; E, I; U, V; 
Contrary pairs are : A, E ; A, U ; 



A PRIMER OF LOGIC 47 

Subcontrary pairs are: I, O; O, Y; 

Subalternate pairs ai'e: A, I; A, V; I, V; I, U; E, O; 

E, U; E, V; U, O. 

In order to show how the new forms fit the ancient 
scheme and as an illustration of mxCthod let us solve the 
array of the s^^llogism. We first observe (see table) that 
A weakens ambiguously to I or V ; that O strengthens am- 
biguously to E or U.* 

Rules : 

1. In any valid mood interchange either premise 
and the conclusion and replace each by its contradictory. 

2. In any valid mood strengthen a premise or weaken 
a conclusion, 

3. In any valid mood convert simply in any form 
but A or O. 

Postulates: 

AAA (1st figure) is a valid mood. 

AUU ( " " ) " " " 

EAE ( *' " ) " " " 

UEA ( " " ) " '» '* »» 

From these rules and postulates will follow sixteen 
valid moods in the 1st figure, twenty in each one of the 
2nd and 3rd figures, and twenty-one in the 4th figure. 

In order to deduce the invalid moods let us assume 
the 

Rules: 

1. In any invalid mood interchange either premise 
and the conclusion and replace each by its contradictory. 

*If X implies y but y does not imply x, then x is said to be a 
strengthened form of y, and y is said to be a weakened form of x. 



48 



A PRIMER OF LOGIC 



2. In any invalid mood weaken a premise or 
strengthen a conclusion. 

3. In any invalid mood convert simply in any form 
but A or O; and 

Postulates: 

AAA (4th figure) is an invalid mood. 

AAO ( " 

AAV (3rd 

AAI (2nd 

AAO (1st 

EEI ( 

EEO ( 

EUA ( 

EUO ( 

UUO ( 

UUV ( 

From these postulates and rules follow the remaining 
(772) invalid moods. 

It will be seen at once that the rules of the old logic 
for the immediate detection of the invalid moods of the 
syllogism no longer hold. To give only one illustration: 
A term may appear distributed in the conclusion of a valid 
mood and be undistributed in the premise. 



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